Maximum likelihood detection of MPSK bursts with inserted reference symbols

ABSTRACT

A fast algorithm for performing maximum likelihood detection of data symbols transmitted as phases of a carrier signal.

CROSS-REFERENCES TO RELATED APPLICATIONS

This patent application is a continuation-in-part of application Ser.No. 08/847,729 filed Apr. 28, 1997, now U.S. Pat. No. 5,940,446.

BACKGROUND OF THE INVENTION

This invention relates generally to the transmission and detection ofdigital data using analog signals, and more particularly the inventionrelates to the detection of phase shift keying (PSK) encoded digitaldata.

The phase of a carrier signal can be used to encode digital data fortransmission. The number of bits represented by a carrier phase symboldepends on the number of phases M of the carrier in an MPSK data burst.

A prior art approach to the detection of data symbols consists of usinga phase locked loop to lock to the reference symbols and then detectingthe data symbols using the phase reference out of the loop. A relatedapproach is to use both reference symbols and remodulated data symbolsto obtain a loop phase reference. These approaches are well known.

Another approach is to form a phase reference using a filteringoperation on the reference symbols, often called pilot symbol aideddemodulation. This approach is essentially the same as the phase lockedloop approach in the sense that the phase locked loop also performs afiltering operation.

The present invention is concerned with maximum likelihood detection ofdata symbols in an MPSK data burst with inserted reference symbols.

SUMMARY AND DESCRIPTION OF THE DRAWINGS

The present invention presents a fast algorithm to perform maximumlikelihood detection of data symbols. The figures of the drawings (FIGS.1A, 1B, 2, 3A, 3B) illustrate flow diagrams of four embodiments inimplementing algorithm.

DETAILED DESCRIPTION OF THE INVENTION

First consider a specific problem which however has all the essentialfeatures of the general problem. Consider that N data symbols s₁, s₂, .. . s_(N) are transmitted at times 1, 2, . . . N, and that a referencesymbol s_(N+1) is transmitted at time N+1. All N+1 symbols are MPSKsymbols, that is, for k=1, . . . N, s_(k)=e^(jφk), where φk is auniformly distributed random phase taking values in {0,2π/M, . . .2π(M−1)/M}, and for k=N+1, reference symbol s_(N+1) is the MPSK symbole^(j0)=1. The N+1 symbols are transmitted over an AWGN (Additive whiteGaussian noise) channel with unknown phase, modeled by the equation:

r=s _(e) ^(jθ) +n.  (1)

where r, s, and n are N+1 length sequences whose k^(th) components arer_(k), s_(k), and n_(k), respectively, k=1, . . . N+1. Further, n is thenoise sequence of independent noise samples, r is the received sequence,and θ is an unknown channel phase, assumed uniformly distributed on(−π,π].

We now give the maximum likelihood decision rule to recover the data s₁,. . . s_(N). For the moment, first consider the problem where we want torecover s=s₁, . . . s_(N+1), where s_(N+1) is assumed to be unknown. Weknow that the maximum likelihood rule to recover s is the s whichmaximizes p(r|s). From previous work, we know that this is equivalent tofinding the s which maximizes η(s), where: $\begin{matrix}{{\eta (s)} = {{{\sum\limits_{k = 1}^{N + 1}{r_{k}s_{k}^{*}}}}^{2}.}} & (2)\end{matrix}$

In general, there are M solutions to (2). The M solutions only differ bythe fact that any two solutions are a phase shift of one another by somemultiple of 2π/M modulo 2π. Now return to the original problem which isto recover the data s₁ . . . s_(N). The maximum likelihood estimate ofs₁, . . . s_(N) must be the first N components of the unique one of theM solutions of (2) whose s_(N+1) component is e^(j0)=1.

An algorithm to maximize (2) when all s_(k), k=1, . . . N+1 are unknownand differentially encoded is given in K. Mackenthun Jr., “A fastalgorithm for multiple-symbol differential detection of MPSK”, IEEETrans. Commun., vol 42, no. 2/3/4, pp. 1471-1474, February/March/April1994. Therefore to find the maximum likelihood estimate of s₁, . . .s_(N) when s_(N+1) is a reference symbol, we only need to modify thealgorithm for the case when s_(N+1) is known.

The modified algorithm to find the maximum likelihood estimate ŝ₁, . . .ŝ_(N) of s₁, . . . s_(N) is as follows. Let Φ be the phase vector Φ=(φ₁,. . . φ_(N+1)), where all φ_(k) can take arbitrary values, includingφ_(N+1). If |r_(k)|=0, arbitrary choice of s_(k) will maximize (2).Therefore, we may assume with no loss in generality that |r_(k)|>0, k=1,. . . N. For a complex number γ, let arg[γ] be the angle of γ.

Let =(₁, . . . _(N+1)) be the unique Φ for which:

arg[r _(k) e ^(−jψ) _(k)]ε[0,2π/M),

for k=1, . . . N+1. Define z_(k) by:

z _(k) =r _(k) e ^(−j−{tilde over (ψ)}) ^(_(k)) .  (3)

For each k, k=1, . . . N+1, calculate arg[z_(k)]. List the valuesarg[z_(k)] in order, from largest to smallest. Define the function k(i)as giving the subscript k of z_(k) for the i^(th) list position, i=1, .. . N+1. Thus, we have: $\begin{matrix}{0 \leq {\arg \left\lbrack z_{k{({N + 1})}} \right\rbrack} \leq {\arg \left\lbrack z_{k{(N)}} \right\rbrack} \leq \ldots \leq {\arg \left\lbrack z_{k{(1)}} \right\rbrack} < {\frac{2\pi}{M}.}} & (4)\end{matrix}$

For i=1, . . . N+1, let:

g _(i) =zk(i).  (5)

For i satisfying N+1<i≦2(N+1), define:

 g_(i) =e ^(−j2πj2) g _(i−(N+1)).  (6)

Calculate: $\begin{matrix}{{{\sum\limits_{i = q}^{q + N}g_{i}}}^{2},\quad {{{for}\quad q} = 1},{{\ldots \quad N} + 1},} & (7)\end{matrix}$

and select the largest.

Suppose the largest magnitude in (7) occurs for q=q′. We now find thephase vector corresponding to q=q′. Using (3), (5), and (6), with i inthe range of q′≦i≦q′+N, we have:

_(k(i))=_(k(i)),q′≦i≦N+1  (8)

$\begin{matrix}{\left. {{\overset{\overset{\sim}{\sim}}{\phi}}_{k{({i - N})}} = {{\overset{\sim}{\phi}}_{k{({i - {\lbrack{N + 1}\rbrack}})}} + \frac{2\pi}{M}}} \right),{{N + 1} < i \leq {q^{\prime} + {N.}}}} & (9)\end{matrix}$

The evaluation of (8) and (9) gives elements _(k(l)), 1=1, . . . N+1, inorder of subscript value k(1), by arranging the elements _(k(l)), l=1, .. . N+1 in order of subscript value k(l), we form the sequence ₁, ₂, . .. , _(N+1), which is the vector . The maximum likelihood estimate of ŝ₁,. . . ŝ_(N) is now given by ŝ_(k)e^(jk), . . . k=1, . . . N, where_(k)=_(k)−_(N+1), k=1, . . . N.

As discussed in Mackenthun supra, algorithm complexity is essentiallythe complexity of sorting to obtain (4), which is (N+1)log(N+1)operations.

We now expand the specific problem considered earlier to a more generalproblem. Suppose that N data symbols are transmitted followed by Lreference symbols s_(N+1), . . . s_(N+L), where s_(k)=e^(j0)=1 fork=N+1, . . . N+L, and assume the definition of channel model (1) isexpanded so that r, s, and n are N+L length sequences. Then in place of(2) we have: $\begin{matrix}{{\eta (s)} = {{{\sum\limits_{k = 1}^{N + L}{r_{k}s_{k}^{*}}}}^{2}.}} & (10)\end{matrix}$

However, note that (10) can be rewritten as: $\begin{matrix}{{\eta (s)} = {{{\sum\limits_{k = 1}^{N}{r_{k}s_{k}^{*}}} + {r_{N + 1}^{\prime}s_{N + 1}^{*}}}}^{2}} & (11)\end{matrix}$

where r′_(N+1)=r_(N+1)+r_(N+2)+ . . . r_(N+L). But we can apply theprevious modified algorithm exactly to (11) and thereby obtain a maximumlikelihood estimate of the first N data symbols.

Now suppose the L reference symbols can take values other than e^(j0).Since the reference symbols are known to the receiver, we can remodulatethem to e^(j0) and then obtain a result in the form (11), and apply theprevious algorithm. Finally, suppose the L reference symbols arescattered throughout the data. It is clear that we can still obtain aresult in the form (11) and apply the previous algorithm.

If desired, sorting can be avoided at the expense of an increase incomplexity in the following way. Fix j, jε{1, . . . N+1}. For k=1, . . .N+1, form r_(j) ^(*)r_(k) and let g_(j,k) be the remodulation of r_(j)^(*)r_(k) such that g_(j,k)ε{0,2π/M}. Now note that the set in (7) isthe same as the set: $\begin{matrix}{{{\sum\limits_{k = 1}^{N + 1}g_{j,k}}}^{2},\quad {{{for}\quad j} = 1},{{\ldots \quad N} + 1.}} & (12)\end{matrix}$

Thus, sorting has been eliminated but forming the above set requires(N+1)² complex multiplications.

The drawing figures illustrate flow diagrams of four embodiments inimplementing the algorithm. The flow chart of FIGS. 1A, 1B can beimplemented in a DSP chip, ASIC chip, or general purpose computer. Thefirst embodiment takes N MPSK data symbols and one reference symbol, ofvalue e^(j0)=1, as input, and produces a maximum likelihood estimate ofthe N data symbols as output. The complexity of the first embodiment isroughly N log N.

FIG. 2 is a second embodiment of the invention, which also takes N MPSKdata symbols and one reference symbol, of value e^(j0)=1, as input, andproduces a maximum likelihood estimate of the N data symbols as output;however, the second embodiment uses a different implementation ofcomplexity roughly N².

FIGS. 3A and 3B are used with FIGS. 1 and 2 to show differentembodiments of the invention, which also form maximum likelihoodestimates of the N data symbols, but which allow for multiple referencesymbols, of arbitrary MPSK values, inserted among the data symbols atarbitrary positions.

FIRST EMBODIMENT OF THE INVENTION

Consider that N data symbols s₁, s₂, . . . s_(N) are transmitted attimes 1,2 . . . N, and that a reference symbol s_(N+1) is transmitted attime N+1. All N+1 symbols are MPSK symbols, that is, for k=1, . . . N,s_(k)=e_(jφk), where φ_(k) is a uniformly random phase taking values in{0,2π/M, . . . 2π(M−1)/M}, and for k=N+1, reference symbol s_(N+1) isthe MPSK symbol e^(j0)=1. The N+1 symbols are transmitted over an AWGNchannel with unknown phase, modeled by the equation (eqn. 1):

r=se ^(j0) +n  (13)

where r, s, and n are N+1 length sequences whose k^(th) components arer_(k), s_(k), and n_(k), respectively, k=1, . . . N+1. Further, n is thenoise sequence of independent noise samples, r is the received sequence,and θ is an unknown channel phase, assumed uniformly distributed on (−π,π].

We now give the maximum likelihood decision rule to recover the data s₁,. . . s_(N). For the moment, first consider the problem where we want torecover s=s₁, . . . s_(N+1), where s_(N+1) is assumed to be unknown. Weknow that the maximum likelihood rule to recover s is the s whichmaximizes p(r|s). From previous work, we know that this is equivalent tofinding the s which maximizes η(s), where $\begin{matrix}{{\eta (s)} = {{{\sum\limits_{k = 1}^{N + 1}{r_{k}s_{k}^{*}}}}^{2}.}} & (14)\end{matrix}$

In general, there are M solutions to (14). The M solutions only differby the fact that any two solutions are a phase shift of one another bysome multiple of 2π/M modulo 2π. Now return to the original problemwhich is to recover the unknown data s₁, . . . s_(N) withs_(N+1)=e^(j0)=1. The maximum likelihood estimate of s₁, . . . s_(N)must be the first N components of the unique one of the M solutions of(14) whose s_(N+1) component is e^(j0)=1.

An algorithm to maximize (14) when all s_(k), k=1, . . . N+1 are unknownand differentially encoded is known. Therefore, to find the maximumlikelihood estimate of s₁, . . . s_(N) when s_(N+1) is a referencesymbol, we only need to modify the algorithm for the case when s_(N+1)is known.

The modified algorithm to find the maximum likelihood estimate ŝ₁, . . .ŝ_(N) of s₁, . . . s_(N) , and the first embodiment of the presentinvention, is as follows. Define R_(k)=r_(k), k=1, . . . N+1. Refer toFIG. 1. The present invention consists of:

a. an input 100 of R_(k), k=1, . . . N+1, where R_(k), k=1, . . . N, areunknown MPSK data symbols s_(k) plus added white Gaussian noise, andR_(N+1) is a known reference symbol e^(j0)=1 plus added white Gaussiannoise.

b. a phase rotator 110 which finds the angle {tilde over (φ)}_(k),{tilde over (φ)}_(k)ε{0,2π/M, . . . 2π(M−1}, such that

arg[R _(k) e ^(−j{tilde over (φ)}k)]ε[0,2π/M),  (15)

for k=1, . . . N+1, where arg[γ] is the angle of the complex number γ.If R_(k)=0, we may assume that φ_(k)=0. At the output of the phaserotator, we define

z _(k) =R _(k) e ^(−j{tilde over (φ)}k).  (16)

c. a division circuit 120 which forms y_(k)=Im(z_(k))/Re(z_(k)), fork=1, . . . N+1.

d. a sorting operation of circuit 130 which orders y_(k) from largest tosmallest, by the index i, i=1, . . . N+1. Define the function k(i) asgiving the subscript k of y_(k) for the i^(th) list position, i=1, . . .N+1. Thus, we have $\begin{matrix}{0 \leq y_{k{({N + 1})}} \leq y_{k{(N)}} \leq \ldots \leq y_{k{(1)}} < {\frac{2\pi}{M}.}} & (17)\end{matrix}$

e. using the function k(i), a circuit 140 reorders z_(k) by defining

g _(i) =z _(k(i)),  (18)

for i=1, . . . N+1.

f. an addition circuit 150 which forms sums S_(q), q=1, . . . N+1, where$\begin{matrix}{{S_{1} = {\sum\limits_{i = 1}^{N + 1}g_{i}}},} & (19)\end{matrix}$

and

S _(q) =S _(q−1) −g _(q−1) +g _(q−1) e ^(−j2π/M),

for q=1, . . . N+1.

g. a squaring and maximization circuit 160 which finds q′ε{1 . . . N+1}such that

|S _(q)′|² ≧|S _(q)|²,

for q=1, . . . N+1.

h. an addition circuit 170 which forms phases _(k(i)) for i=1, . . .N+1, defined by

_(k(i))=_(k(i)),q′≦i≦N+1  (20) $\begin{matrix}{{{\overset{\overset{\sim}{\sim}}{\phi}}_{k{(i)}} = {{\overset{\sim}{\phi}}_{k{(i)}} + \frac{2\pi}{M}}},{{N + 1} < {i + N + 1} \leq {q^{\prime} + {N.}}}} & (21)\end{matrix}$

i. a circuit 180 which reorders _(k(i)) by subscript value to form ₁, ₂,. . . _(N+1).

j. an addition circuit 190 which forms phases _(m), m=1, . . . N,defined by

_(m)=_(m)−_(N+1).   (22)

k. a final circuit 200 which forms a maximum likelihood estimate ŝ_(m)of s_(m) for m=1, . . . N, where ŝ_(m)=e^(jm).

SECOND EMBODIMENT OF INVENTION

In the second embodiment of the invention, we modify the firstembodiment to eliminate sorting, but the implementation complexityincreases from roughly N log N to N².

Define R_(k)=r_(k), k=1, . . . N+1. Refer to FIG. 2. The secondembodiment of the present invention consists of

a. an input 100 of R_(k), k=1, . . . N, where R_(k), k=1, . . . N, areunknown MPSK data symbols s_(k) plus added white Gaussian noise, andR_(N+1) is a known reference symbol e^(j0)=1 plus added white Gaussiannoise.

b. a set of N+1 parallel phase rotators 102, where the q^(th) phaserotator, q=1, . . . N+1, forms $\begin{matrix}{{w_{qk} = \frac{R_{q}^{*}R_{k}}{R_{q}}},} & (23)\end{matrix}$

for k=1, . . . N+1.

c. a set of N+1 parallel phase rotators 112, where the q^(th) phaserotator, q=1, . . . N+1, finds the angle {tilde over (φ)}_(qk),{tildeover (φ)}_(qk)ε{0,2π/M, . . . 2π(M−1)/M}, such that

arg[w _(qk) e ^(−jφ) ^(_(qk)) ]ε[0,2π0,),  (24)

for k=1, . . . N+1. As before, arg[γ] is the angle of the complex numberγ, and if w_(qk)=0, we may assume that _(qk)=0.

d. a set of N+1 parallel circuits 142, where the q^(th) circuit, q=1, .. . N+1, defines g_(qk) by

g _(qk) =w _(qk) e ^(−j{tilde over (ψ)}) ^(_(qk)) ,  (25)

for k=1, . . . N+1.

e. a set of N+1 parallel circuits 152, where the qth circuit, q′=1, . .. N+1, forms a sum S_(q), $\begin{matrix}{S_{q} = {\sum\limits_{k = 1}^{N + 1}{g_{qk}.}}} & (26)\end{matrix}$

f. a squaring and maximization circuit 160 which finds q′ε{1, . . . N+1}such that

|S _(q′)|² ≧|S _(q)|²,  (27)

for q=1, . . . N+1.

g. an addition circuit 192 which forms phases _(m), m=1, . . . N,defined by

_(m)=_(q′,m)−_(q′,N+1).  (28)

h. a final circuit 200 which forms a maximum likelihood estimate ŝ_(m)of s_(m) for m=1, . . . N, where ŝ_(m)=e_(j{circumflex over (ψ)}m).

THIRD AND FOURTH EMBODIMENTS OF INVENTION

Now suppose that N data symbols are transmitted followed by L referencesymbols s_(N+1), s_(N+L), where s_(k)=e^(j0)=1 for k=N+1, . . . N+L.Suppose the L reference symbols are received over the previously definedadditive white Gaussian channel as r_(N+1), . . . r_(N+L). Then we canstill use the first and second embodiment of the invention to derivemaximum likelihood estimates of s₁, . . . s_(N) if the input box 100 inFIGS. 1A and 2 is replaced by input box 100 shown in FIG. 3A. For thethird embodiment, replace 100 in FIG. 1 with 100 in FIG. 3A; for thefourth embodiment, replace 100 in FIG. 2 with 100 in FIG. 3A.

FIFTH AND SIXTH EMBODIMENTS OF INVENTION

Now suppose that N data symbols are transmitted followed by L referencesymbols s_(n+1), . . . s_(N+L), where the L reference symbols aremodulated to arbitrary but known MPSK values, such that s_(k)=e^(jθ)^(_(k)) for s_(k)=e^(jθ) ^(_(k)) for k=N+1, . . . N+L. Suppose the Lreference symbols are received over the previously defined additivewhite Gaussian channel as r_(N+1), . . . r_(N+L). Then we can still usethe first and second embodiment of the invention to derive maximumlikelihood estimates of s₁, . . . s_(N) if the input box 100 in FIGS. 1and 2 is replaced by input box 100 shown in FIG. 3b, that is, if R_(N+1)is redefined as

R _(N+1) =r _(N+1) e ^(−jΘ) ^(_(N+1)) + . . . r _(N+L) e ^(−jΘ)N+L  (29)

For the fifth embodiment, replace 100 in FIG. 1 with 100 in FIG. 3a; forthe sixth embodiment, replace 100 in FIG. 2 with 100 in FIG. 3a.

SEVENTH AND EIGHTH EMBODIMENTS OF THE INVENTION

Now suppose that N data symbols are transmitted along with L referencesymbols which are modulated to arbitrary but known MPSK values, and thatthe reference symbols are inserted among the data symbols at arbitrarypositions. Frequently, the reference symbols are periodically inserted.It is clear from the assumption of the additive white Gaussian noisechannel that we can reindex the data symbols from 1 to N and reindex thereference symbols from N+1 to N+L, and then use the fifth and sixthembodiment of the invention to obtain a maximum likelihood estimate ofthe data symbols, giving the seventh and eighth embodiments of theinvention, respectively.

While the invention has been described with reference to a specificembodiment, the description is illustrative of the invention and is notto be construed as limiting the invention. Various modifications andapplications may occur to those skilled in the art without departingfrom the true spirit and scope of the invention as defined by theappended claims.

What is claimed is:
 1. A method of maximum likelihood detection of datasymbols in an MPSK data burst comprising the steps of: (a) identifying NMPSK data symbols s₁, s₂, . . . s_(N) at times 1,2, . . . N along withat least one reference symbol s_(N+1) at time N+1, where s_(k=e) ^(jψ)_(k) for k=1, . . . N, and φ_(k) is uniformly distributed random phasetaking values in {0,2π/M, . . . 2π(M−1)/M}, and for k=N+1, referencesymbol s_(N+1) is an MPSK symbol e^(j0=1); (b) transmitting said N MPSKsymbols over an AWGN channel with unknown phase and modeled asr=se^(jθ)+n, where r, s, and n are N+1 length sequences whose k^(th)components are r_(k), s_(k), and n_(k), k=1, . . . N+1; and (c) findings which maximized η(s), where:${{\eta (s)} = {{{\sum\limits_{k = 1}^{N}{r_{k}s_{k}^{*}}} + {r_{N + 1}^{\prime}s_{N + 1}^{*}}}}^{2}},$

(c1) defining Φ as the phase vector Φ=(φ₁, . . . φ_(N+1)) and |r_(k)|>0,k=1, . . . N, and for a complex number of γ, let arg[γ] be the angle ofγ; (c2) let Φ=(φ₁, . . . φ_(N+1)) be the unique Φ for which arg[r _(k) e^(−jφ) ^(_(k)) ]ε[0,2πM) for k=1, . . . N+1 and z _(k) r _(k) e^(−j{tilde over (ψ)}) ^(_(k)) ; (c3) for each k, k=1, . . . N+1,calculate arg(z_(k)), and reorder values from largest to smallest, (c4)define a function k(i) as giving a subscript k of z_(k) for the i^(th)list position, i=1, . . . N+1 whereby:${0 \leq {\arg \left\lbrack z_{k{({N + 1})}} \right\rbrack} \leq {\arg \left\lbrack z_{k{(N)}} \right\rbrack} \leq \ldots \leq {\arg \left\lbrack z_{k{(1)}} \right\rbrack} < \frac{2\pi}{M}};$

(c5) for i=1, . . . N+1, let g _(i) =z _(k(i)), and for i satisfyingN+1<i≦2N+1, define: g _(i) =e ^(−j2π/M) g _(i−(N+1)); and (c6)calculate:${{\sum\limits_{i = q}^{q + N}g_{i}}}^{2},\quad {{{for}\quad q} = 1},{{{\ldots \quad N} + 1};\quad {and}}$

(c7) select the largest value in step (c6).
 2. A method of maximumlikelihood detection of data symbols in an MPSK data burst comprisingthe steps of: (a) identifying N MPSK data symbols s₁,s₂, . . . s_(N) attimes 1,2, . . . N along with at least one reference symbol s_(N+1) attime N+1, where s_(k)=e^(jψk) for k=1, . . . N, and φ_(k) is uniformlydistributed random phase taking values in {0,2π/M, . . . 2π(M−1)M}, andfor k=N+1, reference symbol s_(N+1) is an MPSK symbol e^(j0)−1; (b)transmitting said N MPSK symbols over an AWGN channel with unknown phaseand modeled as r=se^(jψ)+n, where r, s, and n are N+1 length sequenceswhose k^(th) components are r_(k) , s_(k) , and n_(k) , k=1, . . . N+1;and (c) finding s which maximized η(s), where:${{\eta (s)} = {{{\sum\limits_{k = 1}^{N}{r_{k}s_{k}^{*}}} + {r_{N + 1}^{\prime}s_{N + 1}^{*}}}}^{2}},$

where R_(N+1)=r_(N+1)+r_(N+2)+ . . . r_(N+L), and L=number of referencesymbols, where step (c) is implemented by: (c1) a division circuit 120which forms y_(k)=Im(z_(k))/Re(z_(k)), for k=1, . . . N+1, (c2) asorting operation of circuit 130 which orders y_(k) from largest tosmallest, define the function k(i) as giving the subscript k of y_(k)for the i^(th) list position, i=1, . . . N+1. Thus, we have$0 \leq y_{k{({N + 1})}} \leq y_{k{(n)}} \leq \ldots \leq y_{k{(1)}} < {\frac{2\quad \pi}{M}.}$

(c3) using the function k(i), a reordering circuit which reorders z_(k)by defining g _(i) =z _(k(i)), for i=1, . . . N+1, (c4) an additioncircuit which forms sums S_(q), q=1, . . . N+1, where${S_{1} = {\sum\limits_{i = 1}^{N + 1}g_{i}}},$

and S _(q) =S _(q−1) −g _(q−1) +g _(q−1) e ^(−j2π/M), for q=1, . . .N+1, (c5) a squaring and maximization circuit which finds q′ε{1m . . .B+1} such that  |S _(q′)|² ≧S _(q)|^(2.) for q=1, . . . N+1, (c6) anaddition circuit which forms phases _(k(i)) for i=1, . . . N+1, definedby _(k(i))=_(k(i)) ,q′ ^(i) ≦i≦N+1${{\overset{\overset{\sim}{\sim}}{\varphi}}_{k{(i)}} = {{\overset{\sim}{\varphi}}_{k{(i)}} + \frac{2\quad \pi}{M}}},{{N + 1} < {i + N + 1} \leq {q^{\prime} + {N.}}}$

(c7) a circuit which reorders _(k(i)) by subscript value to form ₁, ₂, .. . _(N+1). (c8) an addition circuit which forms phases _(m), m=1, . . .N, defined by _(m)=_(m)−_(N+1). (c9) a final circuit which forms amaximum likelihood estimate ŝ_(m) of s_(m) for m=1, . . . N, whereŝ_(m)e^(j{tilde over (φ)}m).
 3. A method of maximum likelihood detectionof data symbols in an MPSK data burst comprising the steps of: (a)identifying N MPSK data symbols s₁, s₂, . . . s_(N) at times 1,2, . . .N along with at least one reference symbol s_(N+1) at time N+1, wheres_(k)=e^(jψk) for k=1, . . . N, and φ_(k) is uniformly distributedrandom phase taking values in {0,2π/M, . . . 2π(M−1)/M}, and for k=N+1,reference symbol s_(N+1) is an MPSK symbol e^(jφ−1); (b) transmittingsaid N MPSK symbols over an AWGN channel with unknown phase and modeledas r=se^(jθ)+n, where r, s, and n are N+1 length sequences whose k^(th)components are r_(k) , s_(k) , and n_(k), k=1, . . . N+1; and (c)finding s which maximized η(s), where:${{\eta (s)} = {{{\sum\limits_{k = 1}^{N}{r_{k}s_{k}^{*}}} + {r_{N + 1}^{\prime}s_{N + 1}^{*}}}}^{2}},$

where R_(N+1)=r_(N+1)+r_(N+2)+ . . . r_(N+L), and L=number of referencesymbols, where step (c) is implemented by: (c1) a set of N+1 parallelphase rotators, where the q^(th) phase rotator, q=1, . . . N+1, forms${w_{qk} = \frac{R_{q}^{*}R_{k}}{R_{q}}},$

for k=1, . . . N+1, (c2) a set of N+1 parallel phase rotators, where theq^(th) phase rotator, q=1, . . . N+1, finds the angle {tilde over(φ)}_(qk),{tilde over (φ)}_(qk)ε{0,2π/M, . . . 2π(M−1)/M}, such thatarg[w _(qk) e ^(−j{tilde over (φ)}) ^(_(qk)) ^(])ε[0,2π0,2, for k=1, . .. N+1, As before, arg[γ] is the angle of the complex number γ, and ifw_(qk)=0, we may assume that φ_(qk)=0, (c3) a set of N+1 parallelcircuits, where the q^(th) circuit, q=1, . . . N+1, defines g_(qk) by g_(qk) =w _(qk) e ^(−j{tilde over (φ)}) ^(_(qk)) , for k=1, . . . N+1,(c4) a set of N+1 parallel circuits, where the q^(th) circuit, q=1, . .. N+1, forms a sum S_(q),$S_{q} = {\sum\limits_{k = 1}^{N + 1}{g_{qk}.}}$

(c5) a squaring and maximization circuit which finds q′ε{1, . . . N+1}such that |S _(q′)|² ≧|S _(q)|², for q=1, . . . N+1, (c6) an additioncircuit which forms phases {tilde over (ψ)}_(m),m=1, . . . N, defined byφ_(m)=φ_(q′,m)−φ_(q′,N+1). (c7) a final circuit which forms a maximumlikelihood estimate ŝ_(m) of s_(m) for m=1, . . . N, whereŝ=e^(j{circumflex over (ψ)}m).